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Extended deformation functors

Marco Manetti∗

Universita di Roma “La Sapienza”, Italy

October 15, 2018

Abstract

We introduce a precise notion, in terms of some Schlessinger’s type conditions, ofextended deformation functors which is compatible with most of recent ideas in theDerived Deformation Theory (DDT) program and with geometric examples. With thisnotion we develop the (extended) analogue of Schlessinger and obstruction theories.The inverse mapping theorem holds for natural transformations of extended deformationfunctors and all such functors with finite dimensional tangent space are prorepresentablein the homotopy category. Finally we prove that the primary obstruction map inducesa structure of graded Lie algebra on the tangent space.

Mathematics Subject Classification (1991): 13D10, 14B10, 14D15.

Introduction

This paper is devoted to the study of some foundations in the derived deformation theory(DDT) program, see [4], [1].

One of the main aspects of DDT program is the Hidden Smoothness Philosophy: this isbased on the idea that, in characteristic 0, every “reasonable” deformation problem give risenaturally to an extended “quasismooth”1 differential Z-graded moduli space MZ such thatthe ordinary moduli space M is just the truncation in degree 0 of MZ. Roughly speaking adifferential Z-graded space is a ringed space whose structure sheaf takes value in differentialZ-graded connected algebras; see [4], [11] for details. In its first written reference [15], theDDT program was proposed as a possible way to define virtual tangent bundle and virtualfundamental class on moduli spaces, while more recent applications of extended moduli spaces[1], [2], concern the understanding of mirror symmetry for Calabi-Yau manifolds.

The hidden smoothness philosophy applied to infinitesimal deformation theory shouldimplies that, over a field of characteristic 0, every reasonable deformation functor is thetruncation of a suitable quasismooth extended deformation functor. Formally a functor Ffrom the category ArtK of local Artinian K -algebras with residue field K to the category ofsets is called a deformation functor if it satisfies the (slightly modified, see the discussion in[6], [7] and references therein) Schlessinger’s conditions

1) F (K ) = {0}.

2) F (A×K B) = F (A)× F (B).

3) The map F (A ×C B) → F (A) ×F (C) F (B) is surjective for every pair of morphismα : A→ C, β : B → C with α surjective.

∗partially supported by Italian MURST program ’Spazi di moduli e teoria delle rappresentazioni’. Memberof GNSAGA of CNR.

1Some authors prefer to say “smooth in an appropriate sense”

1

and the term “reasonable” means essentially that F arises from a geometric deformationproblem.

By an extended deformation functor we understand a set valued functor F defined on acategory C containing ArtK as a full subcategory and such that F satisfies some extendedSchlessinger’s conditions.

As far as we known there are in literature two main notions of extended deformationfunctors.

The first ([1], [2]) considers C as the category of Z-graded local Artinian K -algebras withresidue field K and the extended Schlessinger’s conditions are nothing else than the trivialextension of 1), 2) and 3). This approach does not present any additional difficulty withrespect to the classical case but works well only when the associated extended moduli spaceis smooth in the (strong) sense of [1, 2.2].

The second approach ([13],[4]), takes C as the category of differential Z-graded localArtinian K -algebras with residue field K that are homotopy equivalent to algebras concen-trated in nonpositive degrees and the generalized Schlessinger condition are the above 1), 2),3) together with the following:

4) F sends quasiisomorphisms in C into invertible maps in Set.

This fourth condition is the principal responsible of some hidden smoothness phenomenaof extended deformation functors. This approach works usually quite well for the extensionof (classical) prorepresentable deformation functors, e.g. the Hilbert functor, but fails to begood in more general situations (see the discussion in [16, 3.3]).

In this paper we propose a notion of extended2 deformation functor which contains theprevious two as specializations. In doing that we always keep in mind two general ideas: thefirst is taken from [16], while the second is nowadays standard.

a) C is the category of all differential Z-graded associative (graded)-commutative local Ar-tinian K -algebras with residue field K .

b) Over a field of characteristic 0, every reasonable deformation problem is governed by adifferential graded Lie algebra; therefore the functor of Maurer-Cartan solutions of adifferential graded Lie algebra modulo gauge action must be considered as the basicexample of deformation functor.

Some easy examples show that the above condition 4) is not compatible with a) and b);this is one of the motivation of this paper and forces us to give a more technical definition ofdeformation functor, see 2.1. The paper goes as follows:

In Section 1 we recall some definitions from rational homotopy theory and we fix thenotation for the rest of the paper.

In Section 2 we propose a precise definition of deformation functors in terms of someSchlessinger’s type conditions and we show that our notion is compatible with the aboveideas a) and b). As in the classical case, every deformation functor F has a tangent spaceTF [1] which is a Z-graded vector space. For notational convenience we always replace localArtinian rings with their maximal ideals; this means that, in our description,C is the categoryof finite dimensional associative commutative Z-graded nilpotent differential K -algebras.

In Section 3 we show that the inverse function theorem holds for morphisms of deformationfunctors (Cor. 3.3); this implies in particular a conceptually easier proof of the well knownfact that every quasiisomorphism of differential graded Lie algebras induces an isomorphismsof deformation functors. We also prove that every deformation functor has a natural completeobstruction theory with obstruction space TF [2] (by definition TF [2] is a graded vector spacewith TF [2]i = TF [1]i+1).

2From now on we always omit the term ‘extended’ before ‘deformation functor’

2

In Section 4 we generalize the main result of [23]; every deformation functor F inducesnaturally a functor [F ] defined in the homotopy categoryK(C) (see 1.2). Theorem 4.5 assertsthat, if the tangent space TF [1], is finite dimensional then [F ] is prorepresented by a pronilpo-tent differential algebra (R, d), where R is the inverse limit of R/Rn+1 = ⊕n

i=1(⊙iTF [1]∨)

and d is a differential of degree +1.In Section 5 we will deal with extended deformation functors in the set-up of dg-coalgebras

and we introduce the deformation functors of a L∞-algebra.In Section 6 we will prove the existence of a natural graded Lie algebra structure over the

(shifted) tangent space of a deformation functor.In Section 7 we give a necessary and sufficient condition, always satisfied in concrete

examples, for a deformation functor to be isomorphic to the deformation functor of a L∞-algebra.

This paper owes its existence to the participation of the author to the “Deformationquantization seminar”, held in Scuola Normale Superiore di Pisa during the academic year1998-99. It is a pleasure to thank here E. Arbarello, G. Bini, A. Canonaco, P. de Bartolomeis,F. de Vita, D. Fiorenza, G. Gaiffi, M. Grassi, M. Polito and R. Silvotti for useful and stimu-lating discussions.

Notation

We will always work over a fixed field K of characteristic 0. All vector spaces, linear maps,algebras, tensor products, derivations etc.. are understood of being over K , unless otherwisespecified.

By Σn we denote the symmetric group of permutations of {1, . . . , n}. By S(p, q) ⊂ Σp+q

we denote the set of unshuffles of type (p, q); by definition σ ∈ S(p, q) if and only ifσ(i) < σ(i + 1) for every i 6= p. The unshuffles are a set of representative for the cosetsof Σp × Σq inside Σp+q.

1 Differential graded algebras and homotopy

We denote by: Set the category of sets in a fixed universe; G the category of Z-graded vectorspaces. If V is a graded vector space and v ∈ V , v 6= 0, is a homogeneous element, we denoteby v ∈ Z its degree.DG denotes the category of complexes of vector spaces, also called differential graded vectorspaces. Every object in DG can be considered as a pair3 (V, d), where V is an object inG and the differential d : V → V is a linear map such that d(V i) ⊂ V i+1 and d2 = 0. Wewill consider G as the full subcategory of DG whose objects are the complexes with zerodifferential.

For every complex of vector spaces V ∈ DG we denote, as usual, by Z∗(V ), B∗(V ) andH∗(V ) the cocycles, coboundaries and cohomology of V .

The tensor product in the category DG is defined in the standard way

(V ⊗W )i = ⊕j∈ZVj ⊗W i−j , d(v ⊗ w) = dv ⊗ w + (−1)vv ⊗ dw

The twisting map T : V ⊗W → W ⊗ V is defined by T (x ⊗ y) = (−1)x yy ⊗ x and thenextended by linearity; T is a morphism of complexes and define a Z/2Z-action on V ⊗ V .

3For notational convenience, when the risk of confusion is remote, we prefer to write V for the complexomitting the differential; the same will be made for all algebraic structures builded over a graded vector space.

3

This action extends naturally to an action of the symmetric group Σn on ⊗nV .The Koszul sign ǫ(σ; v1, . . . , vn) = ±1, σ ∈ Σn, is defined by

σ(v1 ⊗ . . .⊗ vn) = ǫ(σ; v1, . . . , vn)(vσ(1) ⊗ . . .⊗ vσ(n)).

In the next formulas we simply write ǫ(σ) instead of ǫ(σ; v1, . . . , vn) when there is no ambi-guity about v1, . . . , vn ∈ V .The symmetric power ⊙nV is the quotient of ⊗nV by the subspace generated by the elementsv1 ⊗ . . .⊗ vn − σ(v1 ⊗ . . .⊗ vn).

Given an integer n, the shift4 functor [n] : DG → DG is defined by setting V [n] =K [n]⊗ V , f [n] = IdK [n] ⊗ f , where

K [n]i =

{

K if i + n = 00 otherwise

Given v ∈ V we also denote by v[n] = 1K [n] ⊗ v ∈ V [n], where 1K [n] is the unity of K

shifted in degree −n; note that v[n] = v − n.More informally, the complex V [n] is the complex V with indices shifted by n and differ-

ential multiplied by (−1)n. Note that HomDG(V,W [n]) are the linear maps f : V →W suchthat f(Vi) ⊂Wi+n for every i and fdV = dW [n]f = (−1)ndW f .

A commutative, associative graded algebra is the data of a graded vector space A ∈ Gtogether an associative, linear multiplication map : A⊗A−→A such that AiAj ⊂ Ai+j and

ab = (−1)a bba.We denote by GA the category of (commutative, associative) graded algebras. A graded

algebra A is called nilpotent if An = 0 for n >> 0; clearly every nilpotent algebra is withoutunit.

A commutative associative differential graded algebra (dg-algebra for short) is an object(A, d) ∈ DG such that A is a graded algebras and d is a derivation of degree 1. This meansthat d satisfies the (graded) Leibnitz rule d(ab) = d(a)b + (−1)aad(b). If A has a unit 1, weassume moreover that 1 6∈ d(A).

We denote by DGA the category of dg-algebras. In the sequel of the paper we considerDG and GA as the full subcategories of DGA whose object are respectively the dg-algebraswith trivial multiplication and with trivial differential. We also denote by NDGA the fullsubcategory of nilpotent dg-algebras and by C the full subcategory of nilpotent dg-algebraswhich are finite dimensional as K -vector space. Note that in our convention DG∩C denotesthe dg-algebras A ∈ C with trivial multiplication.

Example 1.1. If A ∈ C, then K ⊕ A is a local Artinian ring; conversely it is easy to seethat every local Artinian dg-algebra with unit and residue field K has the form K ⊕ A forsome A ∈ C

A module over a dg-algebra A is the data of a complexM ∈ DG together two associativemultiplication maps A ⊗M → M (left multiplication), M ⊗ A → M (right multiplication)which are morphisms in DG commuting with the twisting map. This means that:

• am = (−1)amma for every homogeneous a ∈ A, m ∈M .

• dM (am) = dA(a)m+ (−1)aadM (m), dM (ma) = dM (m)a+ (−1)mmdA(a).

If A has a unit 1 we also assume that 1 ·m = m · 1 = m.If M is an A-module and n ∈ Z then there is a natural structure of A-module over M [n]

having the same right multiplication and the left multiplication induced via the twisting mapT .

4Here we follows the notation of [16]; in [22] the complex V [−n] is called the ‘n-fold suspension of V ’

4

A derivation of a dg-algebra A into an A-moduleM is a morphism of graded vector spacesh : A → M which satisfy the Leibnitz rule h(ab) = h(a)b + ah(b). We denote by Der(A,M)the A0-module of derivations h : A → M . Notice that the differential of A is an element ofDer(A,A[1]) and that if A2 = AM = 0 then Der(A,M) is the space of morphisms in thecategory G from A to M .

In the above set-up it is also defined an A-module Der∗(A,M) ∈ DG with Dern(A,M) =Der(A,M [n]), the obvious left multiplication and differential

δ : Der(A,M [n]) → Der(A,M [n+ 1]); δ(h) = dM [n]h− hdA

Given a morphism of A-modules f : M → A such that f(M)M = 0 (in most applicationsM will be a square-zero ideal of the dg-algebra A and f the inclusion) we define the mappingcone as the dg-algebra C = A ⊕M [1] with the product (a,m)(b, n) = (ab, an +mb) (notethat, as a graded algebra, C is the trivial extension of A by M [1]) and differential

dC =

(

dA f0 dM [1]

)

: A⊕M [1] → A[1]⊕M [2]

The reader must be careful here: the product in C is defined using the A-module structure ofM [1]. We left to the reader the easy verification that the mapping cone C is a dg-algebra, theinclusion A→ C is a morphism of dg-algebras and the projection C →M [1] is a derivation.

Conversely, given a derivation h : B → N we define the derived inverse mapping cone asthe dg-algebra D = B ⊕N [−1] with product (a,m)(b, n) = (ab, an+mb) and differential

dD =

(

dB 0h dN [−1]

)

: B ⊕N [−1] → B[1]⊕N

Here the projection D → B is a morphism of dg-algebras and the inclusion N [−1] → Dis a morphism of D-modules.

We denote by K [t1, . . . , tn, dt1, . . . , dtn] the dg-algebra of polynomial differential formson the affine space A

n with the de Rham differential. We have K [t, dt] = K [t]⊕K [t]dt and

K [t1, . . . , tn, dt1, . . . , dtn] = ⊗i=1

nK [ti, dti]

Since K has characteristic 0, it is immediate to see that H∗(K [t, dt]) = K [0] and then byKunneth formula H∗(K [t1, . . . , tn, dt1, . . . , dtn]) = K [0].Note that for every dg-algebra A and every s = (s1, . . . , sn) ∈ K

n we have an evaluationmorphism es : A⊗K [t1, . . . , tn, dt1, . . . , dtn] → A defined by

es(a⊗ p(t1, . . . , tn, dt1, . . . , dtn)) = p(s1, . . . , sn, 0, . . . , 0)a

For every dg-algebra A we denote A[t, dt] = A ⊗ K [t, dt]; if A is nilpotent then A[t, dt]is still nilpotent. In the next sections the obvious fact that A[t, dt] does not belong to C forevery A 6= 0 will cause some problems whose solution is the introduction, for every positivereal number ǫ > 0 of the dg-subalgebra

A[t, dt]ǫ = A⊕⊕n>0A⌈nǫ⌉ ⊗ (K tn ⊕K tn−1dt) ⊂ A[t, dt]

It is clear that if A ∈ C then A[t, dt]ǫ ∈ C for every ǫ > 0 and A[t, dt] is the union of allA[t, dt]ǫ, ǫ > 0.

Definition 1.2. Given two morphisms of dg-algebras f, g : A → B, a homotopy between fand g is a morphism H : A→ B[t, dt] such that H0 := e0 ◦H = f , H1 := e1 ◦H = g (cf. [12,p. 120]).We denote by [A,B] the quotient of HomDGA(A,B) by the equivalence relation generated

5

by homotopy. Since composition respects homotopy equivalence we can also consider the ho-motopy categories K(DGA), K(NDGA) and K(C); it is convenient to think of K(DGA)as the category with the same objects of DGA and with morphisms Mor(A,B) = [A,B].A morphism in DGA (resp.: NDGA, C) is called a homotopy equivalence if it gives anisomorphism in the corresponding homotopy category.

Two homotopic morphisms induce the same morphism in homology, see [12, p. 120]. IfA,B ∈ DG, then two morphisms f, g : A → B are homotopic in the sense of 1.2 if and onlyif f is homotopic to g in the usual sense. In particular every acyclic complex is contractibleas a dg-algebra.

Definition 1.3. An extension in NDGA is a short exact sequence

0−→I−→Aα

−→B−→0 (1)

such that α is a morphism in NDGA and I is an ideal of A such that I2 = 0. The extension(1) is called small if AI = 0, it is called acyclic small if it is small and I is an acyclic complex,or equivalently if α is a quasiisomorphism.

The easy proof of the following facts is left to the reader:

• Every surjective morphism Aα

−→B in the category C is the composition of a finitenumber of small extensions.

• If Aα

−→B is a surjective quasiisomorphism in C and Ai = 0 for every i > 0 then α isthe composition of a finite number of acyclic small extensions. This is generally false ifAi 6= 0 for some i > 0.

• For every morphism α : A → B in C there exists a surjective homotopy equivalenceγ : C → A in C such that αγ is homotopic to a surjective map. If moreover Ai = Bi = 0for every i > 0 and α : H0(A) → H0(B) is surjective then it is possible to choose Csuch that Ci = 0 for every i > 0.

• Let F : K(C) → Set be a functor such that F (α) is bijective for every acyclic smallextension α; then for every A,B ∈ C with Ai = Bi = 0 for every i > 0 and everyquasiisomorphism γ : A→ B the map F (γ) is bijective.

2 Extended deformation functors

Definition 2.1. A covariant functor F : C → Set is called a predeformation functor if thefollowing conditions are satisfied:

1. F (0) = {0} is the one-point set.

2. (Generalized Schlessinger’s conditions): For every pair of morphisms α : A→ C, β : B →C in C consider the map

η : F (A×C B) → F (A) ×F (C) F (B)

Then:

(a) η is surjective when α is surjective.

(b) η is bijective when α is surjective and C ∈ DG ∩C is acyclic.

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3. (Quasismoothness): For every acyclic small extension

0−→I−→A−→B−→0

the induced map ρ : F (A) → F (B) is surjective.

A predeformation functor F is called a deformation functor if the map ρ defined in 3 isbijective.

The predeformation functors (resp.: deformation functors) together their natural trans-formations form a category which we denote by PreDef (resp.: Def). Note that definition2.1 also makes sense for covariant functors F : NDGA → Set.

Lemma 2.2. For a covariant functor F : C → Set with F (0) = {0} it is sufficient to checkcondition 2b of Definition 2.1 when C = 0 and when B = 0 separately.

Proof Follows immediately from the equality

A×C B = (A×B)×C 0

where Aα

−→C, Bβ

−→C are as in 2b of 2.1 and the fibred product on the right comes from themorphism A×B → C, (a, b) → α(a) − β(b).

Proposition 2.3. Let V be a graded vector space and let (R, d) be a differential algebra suchthat, as a graded algebra, R is the inverse limit of R/Rn+1 = ⊕n

i=1(⊙iV ). Then the functor

hR : NDGA → Set,

hR(A) = HomDGA(R/Rn, A), n >> 0

is a predeformation functor.

Proof We have F (0) = {0} and F (A ×C B) = F (A) ×F (C) F (B); the only nontrivialcondition to check is the surjectivity of F (A) → F (B) for every acyclic small extension0 → I → A

p−→B → 0. For simplicity we prove this in the particular case d(R) ⊂ R2; the

proof in the general case is essentially the same but more messy.Let φ : R/Rn → B be a morphism of dg-algebras, {vi} ⊂ V a homogeneous basis and

denote bi = φ(vi). For every set of liftings ai ∈ A, p(ai) = bi, ai = bi, there exists a uniquemorphism of graded algebras ψ : R/Rn+1 → A such that ψ(vi) = ai; we need to prove thatit is possible to choose the liftings such that ψ(dvi) = dai for every i.

We first note that, since AI = 0 and ψ(dvi) − dai ∈ I, the restriction ψ : R2/Rn+1 → Ais a morphism of dg-algebras independent from the choice of the liftings {ai}. In particularfor every i, d(ψ(dvi) − dai) = 0 and, being I acyclic, there exist si ∈ I, si = ai, such thatψ(dvi)− dai = dsi. It is now sufficient to change ai with ai + si in order to transform ψ intoa morphism of differential graded algebras.

Definition 2.4. We shall call the differential algebra R introduced in 2.3 algebraically free(or quasismooth) complete differential algebra. We also say that it is complete semifree (orsemismooth) if there exists a filtration 0 = H0 ⊂ H1 ⊂ .... ⊂ V of graded vector spaces suchthat ∪∞i=0H

i = V and d(Hi+1) ⊂ lim←

⊕nj=1 (⊙

jHi).

We shall say in addition that R is minimal if d(R) ⊂ R2. Note that, as a graded vector space,V = R/R2.

The terms complete in 2.4 is because R is complete for the R-adic topology.

Examples 2.5.

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1. If R is a complete semifree algebra, it is easy to see that for every surjective quasiiso-morphism A→ B of nilpotent dg-algebras the morphism hR(A) → hR(B) is surjective.This property is generally false if R is assumed algebraically free.

2. Let (R, d) be a complete quasismooth differential algebra, V = R/R2, then R is semifreeif either Vi = 0 for every i > 0 or R is minimal, V1 = 0 and Vj = 0 for every j < 0.In particular the maximal ideal of the algebra of functions of a smooth formal pointeddg-scheme (see [4]) is a complete semifree algebra.

Lemma 2.6. For a predeformation functor F : C → Set the following conditions are equiv-alent:

i) F is a deformation functor.

ii) F induces a functor [F ] : K(C) → Set.

iii) If I ∈ DG ∩C is acyclic then F (I) = {0}.

Proof i)⇒ ii) Let H : A→ B[t, dt] be a homotopy, we need to prove that H0 and H1 inducethe same morphism from F (A) to F (B). Since A is finite-dimensional there exists ǫ > 0sufficiently small such that H : A → B[t, dt]ǫ; now the evaluation map e0 : B[t, dt]ǫ → B isa finite composition of acyclic small extensions and then, since F is a deformation functorF (B[t, dt]ǫ) = F (B); For every a ∈ F (A) we have H(a) = iH0(a), where i : B → B[t, dt]ǫ isthe inclusion and then H1(a) = e1H(a) = e1iH0(a) = H0(a).ii)⇒ iii) In the homotopy category every acyclic complex is isomorphic to 0.iii)⇒ i) We need to prove that for every acyclic small extension

0−→I−→Aρ

−→B−→0

the map F (A) → F (B) is injective, this is done by showing that the diagonal map F (A) →F (A)×F (B)F (A) is surjective; in order to prove this it is sufficient to prove that the diagonalmap A → A ×B A induces a surjective map F (A) → F (A ×B A). We have a canonicalisomorphism θ : A× I → A×B A, θ(a, x) = (a, a+x) which sends A×{0} onto the diagonal;since F (A× I) = F (A)× F (I) = F (A) the proof is concluded.

A standard argument in Schlessinger’s theory [23, 2.10] shows that for every predeforma-tion functor F and every A ∈ C ∩ DG there exists a natural structure of vector space onF (A), where the sum and the scalar multiplication are described by the maps

A×A+

−→A ⇒ F (A×A) = F (A)× F (A)+

−→F (A)

s ∈ K , A·s

−→A ⇒ F (A)·s

−→F (A)

For every deformation functor F and every integer i we denote

T iF = F (K [i− 1]), TF = ⊕i∈ZTiF [−i]

Every natural transformation φ : F → G of deformation functors induces linear mapsT iF → T iG and then a morphism of graded vector spaces TF → TG.

Definition 2.7. Given a deformation functor F , the graded vector spaces TF [1] and TF [2]are called respectively tangent and obstruction space of F .

Theorem 2.8. Let F be a predeformation functor, then there exists a deformation functorF+ and a natural transformation η : F → F+ such that for every deformation functor Gand every natural transformation φ : F → G there exists unique a natural transformationψ : F+ → G such that φ = ψη.

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Proof We first define a functorial relation ∼ on the sets F (A), A ∈ C; we set a ∼ b if andonly if there exists ǫ > 0 and x ∈ F (A[t, dt]ǫ) such that e0(x) = a, e1(x) = b. By 2.6 if Fis a deformation functor then a ∼ b if and only if a = b. Therefore if we define F+ as thequotient of F by the equivalence relation generated by ∼ and η as the natural projection,then there exists a unique ψ as in the statement of the theorem. We only need to prove thatF+ is a deformation functor.

Step 1: If C ∈ DG ∩C is acyclic then F+(C) = {0}.Since C is acyclic there exists a homotopy H : C → C[t, dt]ǫ, ǫ ≤ 1, such that H0 = 0,H1 = Id; it is then clear that for every x ∈ F (C) we have x = H1(x) ∼ H0(x) = 0.

Step 2: ∼ is an equivalence relation on F (A) for every A ∈ C.This is essentially standard (see e.g. [12, p. 125]). In view of the inclusion A → A[t, dt]ǫ therelation ∼ is reflexive. The symmetry is proved simply by remarking that the automorphismof dg-algebras

A[t, dt] → A[t, dt]; a⊗ p(t, dt) → a⊗ p(1− t,−dt)

preserves the subalgebras A[t, dt]ǫ for every ǫ > 0.Consider now ǫ > 0 and x ∈ F (A[t, dt]ǫ), y ∈ F (A[s, ds]ǫ) such that e0(x) = e0(y); we

need to prove that e1(x) ∼ e1(y).Write K [t, s, dt, ds] = ⊕n≥0S

n, where Sn is the n-th symmetric power of the acycliccomplex K t ⊕ K s

d−→K dt ⊕ K ds and define A[t, s, dt, ds]ǫ = A ⊕ ⊕n>0A

⌈nǫ⌉ ⊗ Sn. Thereexists a commutative diagram

A[t, s, dt, ds]ǫt=0−→ A[s, ds]ǫ

ys=0

ys=0

A[t, dt]ǫt=0−→ A

The kernel of the surjective morphism

A[t, s, dt, ds]ǫη

−→A[t, dt]ǫ ×A A[t, dt]ǫ

is equal to ⊕n>0A⌈nǫ⌉ ⊗ (Sn ∩ I), where I ⊂ K [t, s, dt, ds] is the homogeneous differential

ideal generated by st, sdt, tds, dtds. Since I ∩ Sn is acyclic for every n > 0, the morphism ηis a finite composition of acyclic small extensions.

Let ξ ∈ F (A[t, s, dt, ds]ǫ) be a lifting of (x, y) and let z ∈ F (A[u, du]ǫ) be the image of ξunder the morphism

A[t, s, dt, ds]ǫ → A[u, du]ǫ, t→ 1− u, s→ u

The evaluation of z gives e0(z) = e1(x), e1(z) = e1(y).

Step 3: If α : A→ B is surjective then

F (A[t, dt]ǫ)(e0,α)−→ F (A)×F (B) F (B[t, dt]ǫ)

is surjective.

It is not restrictive to assume α a small extension with kernel I. The kernel of (e0, α) isequal to ⊕n>0(A

⌈nǫ⌉∩I)⊗(K tn⊕K tn−1dt) and therefore (e0, α) is an acyclic small extension.

9

Step 4: The functor F+ satisfies 2a of 2.1.

Let a ∈ F (A), b ∈ F (B) be such that α(a) ∼ β(b); by Step 3 there exists a′ ∼ a, a′ ∈ F (A)such that α(a′) = β(b) and then the pair (a′, b) lifts to F (A×C B).

Step 5: The functor F+ satisfies 2b of 2.1.

By 2.2 it is sufficient to verify the condition separately for the cases C = 0 and B = 0.When C = 0 the situation is easy: in fact (A × B)[t, dt]ǫ = A[t, dt]ǫ × B[t, dt]ǫ, F ((A ×B)[t, dt]ǫ) = F (A[t, dt]ǫ) × F (B[t, dt]ǫ) and the relation ∼ over F (A × B) is the product ofthe relations ∼ over F (A) and F (B); this implies that F+(A×B) = F+(A) × F+(B).

Assume now B = 0, then the fibred product D := A×C B is equal to the kernel of α. Weneed to prove that the map F+(D) → F+(A) is injective. Let a0, a1 ∈ F (D) ⊂ F (A) and letx ∈ F (A[t, dt]ǫ) be an element such that ei(x) = ai, i = 0, 1. Denote by x ∈ F (C[t, dt]ǫ) theimage of x by α.

Since C is acyclic there exists a morphism of graded vector spaces σ : C → C[−1] suchthat dσ + σd = Id and we can define a morphism of complexes

h : C → (K s⊕K ds)⊗ C ⊂ C[s, ds]1; h(v) = s⊗ v + ds⊗ σ(v)

The morphism h extends in a natural way to a morphism

h : C[t, dt]ǫ → (K s⊕K ds)⊗ C[t, dt]ǫ

such that for every scalar ζ ∈ K there exists a commutative diagram

C[t, dt]ǫh

−→ (K s⊕K ds)⊗ C[t, dt]ǫ

y

eζ

y

Id⊗eζ

Ch

−→ (K s⊕K ds)⊗ C

Setting z = h(x) we have z|s=1 = x, z|s=0 = z|t=0 = z|t=1 = 0. By step 3 z lifts to anelement z ∈ F (A[t, dt]ǫ[s, ds]1) such that z|s=1 = x; Now the specializations z|s=0, z|t=0, z|t=1

are annihilated by α and therefore give a chain of equivalences in F (D)

a0 = z|s=1,t=0 ∼ z|s=0,t=0 ∼ z|s=0,t=1 ∼ z|s=1,t=1 = a1

proving that a0 ∼ a1 inside F (D).

The combination of Steps 1, 4 and 5 tell us that F+ is a deformation functor.

For later use we point out that in the previous proof (Steps 2,3) we have also showed thefollowing

Lemma 2.9. Let F : C → Set be a predeformation functor and let ν : F → F+ be theprojection onto the associated deformation functor.Then for every surjective morphism A→ B in C, the map

F (A) → F (B)×F+(B) F+(A)

is surjective.

For computational purposes it is often useful the following

Lemma 2.10. Let S be a complex of vector spaces and assume that the functor

DG ∩C → Set; C → Z1(S ⊗ C)

is the restriction of a predeformation functor F . Then for every complex C ∈ DG ∩ C wehave F+(C) = H1(S ⊗ C); in particular T iF+ = Hi(S).

10

Proof Let C ∈ DG ∩ C be a fixed complex and set A = S ⊗ C; since A2 = C2 = 0 forevery ǫ > 0 we have A[t, dt]ǫ = S ⊗ C[t, dt]ǫ.

For every integer i ≥ 2 let pi(t) ∈ K [t] be a fixed monic polynomial of degree i such thatp(0) = p(1) = 0; every element ξ ∈ Z1(A[t, dt]ǫ) = F (C[t, dt]ǫ) can be written as

ξ = a0 + a1t+∑

i≥2

aipi(t) + b1dt+∑

i≥2

bip′i(t)dt

with ai ∈ A1, bi ∈ A0, da0 = 0 and dbi = ai for every i. In particular e1(ξ) − e0(ξ) =a1 = db1 ∈ B1(A). Conversely if a0, a1 ∈ Z1(A) and a1 − a0 = db1 for some b1 ∈ A0 thenaj = ej(a0 + (a1 − a0)t+ b1dt) for j = 0, 1.

Example 2.11. Let R be an algebraically free complete differential algebra;

1. For every complex C we have

hR(C) = HomDG(R/R2, C) = Z0(Hom∗(R/R2,K )⊗ C)

and then T ih+R = Hi−1(Hom∗(R/R2,K )).

2. The functor [R,−] : NDGA → Set is a deformation functor with tangent space

T i[R,−] = Hi−1(Hom∗(R/R2,K )) = Hi−1((R/R2)∨).

The proof of this fact is essentially the same (but more easy) of 2.8 and it is omitted.If the graded vector space R/R2 is finite-dimensional then it is easy to prove directlythat h+R = [R,−]

3. The projection hR → [R,−] factors through a morphism of deformation functors h+R →[R,−] which is an isomorphism on tangent spaces.

All the deformation functors are quasismooth by definition; a stronger notion of smooth-ness is given by the natural generalization of the classical notion of smooth morphism.

Definition 2.12. A natural transformation of deformation functors θ : F → G is calledsmooth if, for every morphism A→ B in C, which is surjective in homology the natural map

F (A) → F (B)×G(B) G(A)

is surjective. A deformation functor F is called smooth if the trivial morphism F → 0 issmooth.

Proposition 2.13. Let (S, d) be a minimal complete quasismooth differential algebra. Thenthe deformation functor [S,−] is smooth if and only if d = 0.

Proof If d = 0 it is easy to see that [S,−] is smooth. Conversely assume d 6= 0; let n ≥ 2be the greatest integer such that d(S) ⊂ Sn and let π : S/S2 → V be a finite dimensionalquotient such that the composition H

d−→Sn/Sn+1 → ⊙n(V ) is nonzero.

Let (R, 0) be the complete free differential algebra with R/R2 = V ; we shall see that[S,R/Rn+1] → [S,R/Rn] is not surjective, this implies that [S,−] is not smooth. Since S/Sn

and R/Rn have differential d = 0 we have [S,R/Rn] = HomDGA(S,R/Rn) and therefore itis sufficient to show that the projection π : S → R/Rn does not lift to R/Rn+1. Assume thatthere exists f : S → R/Rn+1 which is a lifting of π; let x ∈ S/S2 be such that π(dx) 6= 0 in Rn,then f(x) = π(x)+α for some α ∈ Rn/Rn+1; in particular the restriction of f to S2 is equalto π. We must therefore have 0 = df(x) = f(dx) = π(dx) 6= 0 giving a contradiction.

The second basic example of deformation functor is the deformation functor of a differ-ential graded Lie algebra; we recall the well known

11

Definition 2.14. A differential graded Lie algebra (DGLA in short terms) is a triple (L, d, [, ])with (L, d) ∈ DG and [, ] : L× L→ L is a bilinear map such that:

• [Li, Lj] ⊂ Li+j.

• [x, y] + (−1)x y[y, x] = 0.

• (Jacoby identity) [x, [y, z]] = [[x, y], z] + (−1)x y[y, [x, z]]

• d[x, y] = [dx, y] + (−1)x[x, dy]

Example 2.15. Let A be a dg-algebra, then the A-module Der∗(A,A) has a natural struc-

ture of DGLA with bracket [δ, τ ] = δ ◦ τ − (−1)δτ τ ◦ δ, where the composition ◦ is made byconsidering δ and τ as linear endomorphism of the K -vector space A.

We shall denote by DGLA the category of differential graded Lie algebras: a morphismof DGLA is simply a morphism of complexes which commutes with brackets.

The Maurer-Cartan elements of a DGLA L are by definition

MC(L) =

{

x ∈ L1

∣

∣

∣

∣

dx+1

2[x, x] = 0

}

Clearly every morphism of differential graded Lie algebras L→ N sends the Maurer-Cartanelements of L into the ones of N .

Given a DGLA L and A ∈ C, the tensor product L⊗A has a natural structure of nilpotentDGLA with

(L ⊗A)i = ⊕j∈ZLj ⊗ Ai−j

d(x⊗ a) = dx⊗ a+ (−1)xx⊗ da

[x⊗ a, y ⊗ b] = (−1)a y[x, y]⊗ ab

Every morphism of DGLA, L→ N and every morphism A→ B in C give a natural commu-tative diagram of morphisms of differential graded Lie algebras

L⊗A −→ N ⊗A

y

y

L⊗B −→ N ⊗B

Definition 2.16. The Maurer-Cartan functor MCL : C → Set of a DGLA L is given by

MCL(A) =MC(L ⊗A)

Lemma 2.17. In the notation above MCL is a predeformation functor; the resulting functorMC : DGLA → PreDef is faithful.

12

Proof It is evident thatMCL(0) = 0 and for every pair of morphisms α : A→ C, β : B → Cin C we have

MCL(A×C B) =MCL(A)×MCL(C) MCL(B)

Let 0−→I−→A−→B−→0 be an acyclic small extension and x ∈MCL(B). Since α is surjec-tive there exists y ∈ (L⊗A)1 such that α(y) = x. Setting

h = dy +1

2[y, y] ∈ (L⊗ I)2

we have

dh =1

2d[y, y] = [dy, y] = [h, y]−

1

2[[y, y], y].

By Jacoby identity [[y, y], y] = 0 and, since AI = 0 also [h, y] = 0; thus dh = 0 and, beingL ⊗ I acyclic by Kunneth formula, there exists s ∈ (L ⊗ I)1 such that ds = h. The elementy − s lifts x and satisfies Maurer-Cartan equation. We have thus proved that MCL is apredeformation functor.

The natural identification Li =MCL((K t⊕K dt)[i]) shows thatMC : DGLA → PreDefis faithful.

Remark 2.18. It is easy to prove that a differential graded Lie algebra can be recovered, upto isomorphism, from its Maurer-Cartan functor.

In order to motivate our definition of deformation functor we point out that, if A→ B is asurjective quasiisomorphism in C, then in general MCL(A) → MCL(B) is not surjective. Asan example take K algebraically closed and L a finite-dimensional non-nilpotent Lie algebra,considered as a DGLA concentrated in degree 0. Fix a ∈ L such that ad(a) : L → L has aneigenvalue λ 6= 0. Up to multiplication of a by −λ−1 we can assume λ = −1. Let V ⊂ L bethe image of ad(a), the linear map Id+ ad(a) : V → V is not surjective and then there existsb ∈ L such that the equation x+ [a, x] + [a, b] = 0 has no solution in L.

Let u, v, w be indeterminates of degree 1 and consider the dg-algebras

B = K u⊕K v, B2 = 0, d = 0

A = K u⊕K v ⊕Kw ⊕K dw, uv = uw = dw, vw = 0

The projection A → B is a quasiisomorphism but the element a ⊗ u + b ⊗ v ∈ MCL(B)cannot lifted to MCL(A). In fact, if there exists ξ = a⊗ u+ b⊗ v + x⊗w ∈MCL(A), then

0 = dξ +1

2[ξ, ξ] = (x + [a, x] + [a, b])⊗ dw

in contradiction with the previous choice of a, b.

To every differential graded Lie algebra (L, d, [, ]) we can associate a new DGLA (Ld, dd, [, ]d)by setting Ld = L⊕K d, where d is considered as an indeterminate of degree 1, dd(a+αd) =d(a) and

[a+ αd, b+ βd]d = [a, b] + αd(b) − (−1)aβd(a)

Note that a ∈MC(L) if and only if a ∈ L1 and [a+ d, a+ d]d = 0.If L0 = L0

d is a nilpotent Lie algebra then the adjoint action L0 × L1d

[,]−→L1 ⊂ L1

d givesa group action exp(L0)× L1

d → L1d preserving the affine hyperplane L1 + αd for every fixed

α ∈ K . As a consequence of Jacoby identity (see [20], [8] for details) this action preserves

13

the quadratic cone {x ∈ L1d | [x, x]d = 0} and then exp(L0) acts, via the above identification,

on the set MC(L).The restriction of the adjoint action toMC(L) is usually called “gauge action”. Note that

if, for a ∈ L0 and b ∈MC(L), we have [a, b+ d]d = [a, b]− da = 0 then eab = b; in particularth set Kb = {[b, c] + dc | c ∈ L−1} is a Lie subalgebra of L0 and exp(Kb) is contained in thestabilizer of b.

For every DGLA L and every A ∈ C we define DefL(A) as the quotient ofMC(L⊗A) bythe gauge action of the group exp((L ⊗ A)0). The gauge action commutes with morphismsin C and with morphisms of differential graded Lie algebras and then the above definitiongives a functor DefL : C → Set called the deformation functors of L: in fact we have

Theorem 2.19. For every differential graded Lie algebra L, DefL is a deformation functorwith T iDefL = Hi(L).

Proof If C ∈ DG ∩ C then L ⊗ C is an abelian DGLA, MCL(C) = Z1(L ⊗ C) andthe gauge action is given by eab = b − da, b ∈ (L ⊗ C)1, a ∈ (L ⊗ C)0. This proves thatDefL(C) = H1(L ⊗ C) and in particular T iDefL = H1(L ⊗K [i− 1]) = Hi(L).

Since DefL is the quotient of a predeformation functor and DefL(I) = 0 for every acyclicI ∈ DG ∩ C, it is sufficient to verify the generalized Schlessinger’s conditions. Let α : A →C, β : B → C morphism in C with α surjective. Assume there are given a ∈ MCL(A),b ∈ MCL(B) such that α(a) and β(b) give the same element in DefL(C); then there existsu ∈ (L⊗C)0 such that β(b) = euα(a). Let v ∈ (L⊗A)0 be a lifting of u, changing if necessarya with its gauge equivalent element eva, we may suppose α(a) = β(b), the pair (a, b) lifts toMCL(A×C B); this proves that the map

DefL(A×C B) → DefL(A) ×DefL(C) DefL(B)

is surjective.If C = 0 then the gauge action exp((L ⊗ (A× B))0)×MCL(A × B) → MCL(A× B) is

the direct product of the gauge actions exp((L⊗A)0)×MCL(A) →MCL(A) and exp((L⊗B)0)×MCL(B) →MCL(B) and therefore DefL(A×B) = DefL(A)×DefL(B).

Finally assume B = 0, C acyclic with C2 = 0 and set D = kerα ≃ A ×C B. Leta1, a2 ∈ MCL(D), u ∈ (L ⊗ A)0 be such that a2 = eua1; we need to prove that there existsv ∈ (L⊗D)0 such that a2 = eva1.

Since α(a1) = α(a2) = 0 and L⊗C is an abelian DGLA we have 0 = eα(u)0 = 0− dα(u)and then there exists h ∈ (L ⊗ A)−1 such that dα(h) = −α(u), u + dh ∈ (L ⊗ D)0. Ifw = [a1, h] + dh then ewa1 = a1, e

uewa1 = eva1 = a2 where v = u ∗ w is determined byBaker-Campbell-Hausdorff formula.We claim that v ∈ L⊗D; in fact v = u ∗w ≡ u+w ≡ u+ dh (mod [L⊗A,L⊗A]) and sinceA2 ⊂ D we have v = u ∗ w ≡ u+ dh ≡ 0 (mod L⊗D).

Corollary 2.20. For every differential graded Lie algebra L the natural projection MCL →DefL induces (by 2.8) a morphism MC+

L → DefL which is an isomorphism on tangent spaces.

3 Obstruction theory and the inverse function theorem

for deformation functors

Theorem 3.1. A morphism of predeformation functors θ : F → G is an isomorphism if andonly if θ : F (A) → G(A) is a bijection for every complex A ∈ C ∩DG.

14

Proof The proof uses the natural generalization to the differential graded case of somestandard techniques in Schlessinger’s theory, cf. [6].

Let θ : F → G be a fixed natural transformation of predeformation functors such thatθ : F (A) → G(A) is bijective for every A ∈ DG ∩C.

Step 1: For every small extension

0−→I−→Aα

−→B−→0

and every b ∈ F (B) we have either α−1(b) = ∅ or θ(α−1(b)) = α−1(θ(b)).As in the classical case [23], there exists an isomorphism of dg-algebras

A× I−→A×B A; (a, t) → (a, a+ t)

and then for every predeformation functor E there exists a natural surjective map

ϑE : E(A)× E(I) = E(A× I) → E(A)×E(B) E(A)

This implies in particular that there exists a natural transitive action of the vector spaceE(I) on the fibres of the map E(A) → E(B). Moreover this action commutes with naturaltransformations of functors.

In our case we have a commutative diagram

F (A)α

−→ F (B)

yθ

yθ

G(A)α

−→ G(B)

and compatible transitive actions of the vector space F (I) = G(I) on the fibres of thehorizontal maps. This proves Step 1.

Step 2: Let Aα

−→V be a surjective morphism in C with V ∈ DG∩C acyclic. Denotingby ι : B = kerα → A the inclusion, then an element b ∈ G(B) lifts to F (B) if and only if ι(b)lifts to F (A).

In fact, by condition 2b of 2.1, the inclusion ι gives bijection F (B) = α−1(0) ⊂ F (A),G(B) = α−1(0) ⊂ G(A). Since F (V ) = G(V ) we have

ιθ(F (B)) = θ(F (A)) ∩ α−1(0) = θ(F (A)) ∩ ι(G(B))

Step 3: Let

0−→Iι

−→Aα

−→B−→0

be a small extension and b ∈ F (B). Then b lifts to F (A) if and only if θ(b) lifts to G(A).The only if part is trivial, let’s prove the if part. Let a ∈ G(A) be a lifting of θ(b) and let

C = A⊕ I[1] be the mapping cone of ι. The projection π : C → B is a small acyclic extensionand then there exists b1 ∈ F (C) which lifts b. By Step 1 we can assume that θ(b1) = j(a),where j : A→ C is the inclusion.

If ob : C → I[1] denotes the projection then ob(θ(b1)) = 0 and since F (I[1]) = G(I[1]) wealso have ob(b1) = 0; by generalized Schlessinger’s conditions b1 lifts to F (A).

15

Step 4: For every A ∈ C the map θ : F (A) → G(A) is surjective.By induction over dimK A we can assume that F (B) goes surjectively onto G(B) for every

proper quotient B = A/I. Let

0−→I−→Aα

−→B−→0

be a small extension with I 6= 0, a ∈ G(A) a fixed element and b ∈ F (B) such that θ(b) =α(a). By Step 3 α−1(b) is not empty and then by Step 1 a ∈ θ(F (A)).

Step 5: Let a ∈ F (A), for every surjective morphism f : A → B in the category C wedefine

SF (a, f) = {ξ ∈ F (A×B A) | ξ → (a, a) ∈ F (A)×F (B) F (A)}

By definition, if f is a small extension and I = ker f then SF (a, f) is naturally isomorphicto the stabilizer of a under the action of F (I) on the fibre f−1(f(a)). It is also clear thatθ(SF (a, f)) ⊂ SG(θ(a), f).

Step 6: For every a ∈ F (A) and every surjective morphism f : A → B the mapθ : SF (a, f) → SG(θ(a), f) is surjective.

This is trivially true if B = 0, we prove the general assertion by induction on dimK B.Let

0−→I−→Bα

−→C−→0

be a small extension with I 6= 0, set g = αf and denote by h : A ×C A → I the surjectivemorphism of dg-algebras defined by h(a1, a2) = f(a1)−f(a2); the kernel of h is A×BA, let’sdenote by ι : A×B A→ A×C A the natural inclusion.

By generalized Schlessinger’s conditions the maps

F (A×B A) → ker(F (A×C A)h

−→F (I)); SF (a, f) → SF (a, g) ∩ kerh

are surjective. Let ξ ∈ SG(θ(a), f) and let η ∈ SF (a, g) such that θ(η) = ι(ξ). Since F (I) =G(I) we have h(η) = 0 and then η lifts to some ξ1 ∈ SF (a, f). Let D = (A×C A)⊕ I[−1] bethe derived inverse mapping cone of h, it is immediate to check that the projection maps

π1 : D → A×C A, (h, π2) : D → I ⊕ I[−1]

are surjective morphisms of dg-algebras, I ⊕ I[−1] is an acyclic complex and the kernel of(h, π2) is exactly A×B A; again by generalized Schlessinger’s conditions F (A×B A) ⊂ F (D),G(A×B A) ⊂ G(D). Moreover there exists a cartesian diagram

(A×B A)× I[−1] −→ A×B A

y

y

ι

Dπ1−→ A×C A

which gives surjective maps

F (A×B A)× F (I[−1])

−→F (A×B A)×F (A×CA) F (D)

G(A×B A)×G(I[−1])

−→G(A ×B A)×G(A×CA) G(D)

This implies the existence of v ∈ G(I[−1]) = F (I[−1]) such that (θ(ξ1), v) = (θ(ξ1), ξ);defining ξ ∈ F (D) by the formula (ξ1, v) = (ξ1, ξ) we get θ(ξ) = ξ and then, by Step 2,ξ ∈ SF (a, f).

16

Step 7: For every A ∈ C the map θ : F (A) → G(A) is injective.This is true by assumption if A2 = 0; if A2 6= 0 we can suppose by induction that there

exists a small extension

0−→Iι

−→Aα

−→B−→0

with I 6= 0 and θ : F (B) → G(B) injective.Let a1, a2 ∈ F (A) be two elements such that θ(a1) = θ(a2); by assumption f(a1) = f(a2)

and then there exists t ∈ F (I) such that ϑF (a1, t) = (a1, a2), ϑG(θ(a1), θ(t)) = (θ(a1), θ(a2))and then θ(t) ∈ SG(θ(a1), α). By Step 6 there exists s ∈ SF (a1, α) such that θ(s) = θ(t) andby injectivity of θ : F (I) → G(I) we get s = t and then a1 = a2.

The result of Theorem 3.1 is particularly useful for morphisms of deformation functors:In fact we have

Lemma 3.2. Let F : C → Set be a deformation functor; for every I ∈ DG ∩C there existsa natural isomorphism

F (I) = ⊕i∈Z

TF [1]i ⊗H−i(I) = ⊕i∈ZT i+1F ⊗H−i(I) = H0(Hom∗(I∨, TF [1])).

Proof Let s : H∗(I) → Z∗(I) be a linear section of the natural projection, then the compo-sition of s with the natural embedding Z∗(I)

ι−→I is unique up to homotopy and its cokernel

is an acyclic complex, therefore it gives a well defined isomorphism F (H∗(I)) → F (I). Thissays that it is not restrictive to prove the lemma for complexes with zero differential. More-over since F commutes with direct sum of complexes we can reduce to consider the casewhen I ≃ K

s[n] is a vector space concentrated in degree −n. Every v ∈ I gives a morphismF (K [n])

v−→F (I) and we can define a natural map T 1+nF ⊗ I → F (I), x ⊗ v → v(x). It is

easy to verify that this map is an isomorphism of vector spaces.

As an immediate consequence we have:

Corollary 3.3. A morphism of deformation functors θ : F → G is an isomorphism if andonly if it gives an isomorphism of tangent spaces θ : TF [1]

≃−→TG[1]

Corollary 3.4. For every differential graded Lie algebra L there exists a natural isomor-phism MC+

L = DefL.

Proof According to 2.8 the projection MCL → DefL induces a natural transformationMC+

L → DefL which is an isomorphism on tangent spaces by 2.20.

Corollary 3.5. Every quasiisomorphism φ : L → N of differential graded Lie algebras in-duces an isomorphism DefL ≃ DefN between deformation functors.

Proof Trivial consequence of 2.19 and 3.3.

Corollary 3.6. For every complete quasismooth dg-algebra R there exists an isomorphismh+R = [R,−].

17

Proof Follows from 2.11 and 3.3. Note that this is a nontrivial result when the gradedvector space R/R2 is infinite dimensional.

The argument used in the proof of 3.1 can be used to show the existence of a com-plete natural (and probably universal, see [6] for the definition) obstruction theory for everydeformation functor F .

Given a small extension

e : 0−→Iι

−→Aα

−→B−→0

we can define an “obstruction map” obe : F (B) → F (I[1]) = H0(TF [2]⊗ I) in the followingway:Let C = A ⊕ I[1] be the mapping cone of the inclusion ι; since the projection C → B is anacyclic small extension, the projection C → I[1] gives a map obe : F (B) = F (C) → F (I[1]).

The obstruction maps satisfy the following properties:

• obe(b) = 0 if and only if b lifts to F (A).

• (naturality) The obstruction maps commute with natural transformation of functors.

• (base change) Given a morphism of small extensions

e : 0 −→ I −→ Aα

−→ B −→ 0

y

y

f

y

y

f

e′ : 0 −→ I ′ −→ A′α′

−→ B′ −→ 0

we have f ◦ obe = obe′ ◦ f : F (B) → F (I ′[1]).

The last two items are straightforward, while the first follows by generalized Schlessinger’sconditions applied to the cartesian diagram

A −→ 0

y

y

C −→ I[1]

Example 3.7. If 0−→I−→J−→L−→0 is a short exact sequence of complexes then the ob-struction map F (L) → F (I[1]) is the tensor product of the identity over TF and the con-necting homomorphism H∗(L) → H∗(I[1]).

Remark 3.8. The above construction also shed light on the obstruction theory for generalextensions. In this case it is necessary to replace obstruction maps with obstruction sections.If

e : 0−→Iι

−→Aα

−→B−→0; I2 = 0

is an extension, C = A⊕ I[1] is the mapping cone of the inclusion and B⊕ I[1] is the trivialextension (i.e. the mapping cone of the zero map I[1]

0−→B) then the obstruction section

obe : F (B) = F (C) → F (B⊕I[1]) is the map induced by the natural projection C → B⊕I[1].The same proof as above shows that b ∈ F (B) lifts to F (A) if and only if obe(b) = j(b), wherej : F (B) → F (B ⊕ I[1]) is the “zero section” induced by the natural inclusion B ⊂ B ⊕ I[1].Clearly if e is a small extension then BI = 0, F (B ⊕ I[1]) = F (B)× F (I[1]) and we recoverthe notion of obstruction maps.

18

Example 3.9. (Primary obstruction maps)Let u, v be indeterminates of degrees −i,−j respectively and F : C → Set a deformationfunctor; let (u, v) ⊂ K [u, v] be the ideal generated by u, v and denote by

Q1ij : T

1+iF × T 1+jF = F (K u⊕Kv) → F (K uv[1]) = T 2+i+jF

the obstruction map associated to the small extension

0−→K uv−→(u, v)

(u2, v2)−→K u⊕Kv−→0.

In the realm of obstruction theory belong the following results which we will use in thenext sections.Consider a generic small extension in C

e : 0−→Iι

−→Aα

−→B−→0

with obstruction map obe : F (B) → F (I[1]). Let’s denote by d0 : A→ A[1] the differential ofA.

For every morphism of dg-algebras φ : B → I[1] the map dφ = d0 + ιφα : A → A[1] is anew differential on A: in fact BI = 0, this implies that ιφα is a derivation and

d2φ = d0ιφα + ιφαd0 = ι(dIφ+ φdB)α = ι(−dI[1]φ+ φdB)α = 0.

Denote by Aφ the graded algebra A with the differential dφ and by

eφ : 0−→Iι

−→Aφα

−→B−→0

the corresponding small extension.

Lemma 3.10. In the notation above obeφ = obe + φ : F (B) → F (I[1]).

Proof Let Aφ ⊕ I[1], A⊕ I[1] be respectively the mapping cones of the inclusions Iι

−→Aφ,I

ι−→A and define the following isomorphism of graded algebras

h =

(

Id 0φα Id

)

: Aφ ⊕ I[1] → A⊕ I[1]

Since(

Id 0φα Id

)(

dφ ι0 dI[1]

)

=

(

d0 ι0 dI[1]

)(

Id 0φα Id

)

the morphism h is also an isomorphism of dg-algebras. There exists a commutative diagram

Aφ ⊕ I[1]h

−→ A⊕ I[1]

yα×Id

yα×Id

B × I[1]l

−→ B × I[1]

l =

(

Id 0φ Id

)

showing that, via the bijections α : F (Aφ ⊕ I[1])≃−→F (B), α : F (A ⊕ I[1])

≃−→F (B) the iso-

morphism h acts trivially on F (B). Therefore the obstruction section Id × obeφ : F (B) →F (B)× F (I[1]) is precisely the composition with l of the obstruction section Id× obe.

Let’s consider two dg-algebras (B, dB) ∈ C, (I, dI) ∈ C ∩DG and a small extension ofnilpotent graded algebras

0−→Iι

−→Aα

−→B−→0

Assume that there exists a lifting of dB to a derivation d ∈ Der(A,A[1]) such thatd|I = dI . Such a lifting is not unique, any two liftings differ by an derivation of the form ιφαfor some derivation φ : B → I[1].

19

Lemma 3.11. In the above notation, the derivation d2 = 12 [d, d] : A→ A[2] may be factored

as d2 = δα, where δ : B → I[2] is a morphism of dg-algebras whose homotopy class [δ] ∈[B, I[2]] is independent from the choice of the lifting. Moreover [δ] = 0 if and only if thereexists a lifting d as above such that d2 = 0.

Proof It is evident that d2(A) ⊂ I[2], d2(I) = 0 and then d2 induces a derivation δ : B →I[2]; since d2 ◦ d = dI ◦ d2 = dI[2] ◦ d

2 we have that δ is also a morphism of dg-algebras and

then gives a morphism of complexes δ : B/B2 → I[2].If d′ is another lifting as above and φ = d′−d, then the derivation φ comes in the obvious

way from a morphism of graded vector spaces φ : B/B2 → I[1]; since δ′− δ = (d+φ)2−d2 =dφ+φd we have that φ is the required homotopy between δ and δ′. If δ : B → I[2] is homotopicto 0 then, being I[2][t, dt]2 = 0, also δ is homotopic to 0 and then there exists a derivationφ : A→ B → B/B2 → I[1] such that d2 = dφ+ φd, (d− φ)2 = 0.

In the above set-up, for every deformation functor F , the map δ : F (B) → F (I[2]) is, apriori, independent from the choice of the lifting of the differential. A posteriori we have thefollowing stronger result:

Proposition 3.12. In the above notation δ(F (B)) = 0; in particular if there exists ξ ∈ F (B)such that the map

[B, I[2]] → F (I[2]); [f ] → f(ξ)

is injective then there exists a dg-algebra structure on A making 0−→Iι

−→Aα

−→B−→0 asmall extension in C.

Proof We have already seen that δ induces a morphism of complexes δ : B/B2 → I[2]. Itis sufficient to construct a commutative diagram in C

Cγ

−→ B

y

y

Vγ

−→ B/B2

such that γ is an acyclic small extension, V 2 = 0 and δγ(H∗(V )) = 0 ⊂ H∗(I[2]).Our solution is to define C = A× I[1] as a graded algebra and γ as the composition of α

with the projection on the first factor. Over C we put the differential

dC =

(

d ι−d2 dI[1]

)

: A× I[1] → A[1]× I[2]

A simple verification shows that dC is a derivation, d2C = 0 and γ is a morphism of dg-algebras. The kernel of γ is the mapping cone of the identity I → I and therefore γ is anacyclic small extension.

Finally we set V = B/B2 × I[1] and the differential dV is forced by dC to be equal to

dV =

(

dB/B2 0

−δ dI[1]

)

Since δγ(b, t) = δ(b) and dV (b, t) = 0 ⇒ δ(b) = dI[1](t) = −dI[2](t) the morphism δγ istrivial in homology.

20

4 Finite deformation functors

The notion of homotopy equivalence of morphisms works well for finitely generated or nilpo-tent dg-algebras but it seems quite restrictive for general differential algebras. In the set-upof projective limits of nilpotent dg-algebras it is useful to introduce a notion which arisesnaturally when we consider projective limits in homotopy categories.

Definition 4.1. Let f, g : S → R be morphisms of differential algebras, with R complete forthe R-adic topology (i.e. R = lim

←R/Rn); we shall say that f is prohomotopy equivalent to

g if the morphisms fn, gn : S → R → R/Rn, composition of f, g with the natural projectionR → R/Rn, are homotopy equivalent for every n.

It is clear that the relation defined in 4.1 is an equivalence relation, if R is nilpotent it isthe same of the usual homotopy equivalence.

If f, g : S → R are prohomotopy equivalent morphisms of quasismooth complete dif-ferential algebras then for every A ∈ C and every φ : R → A there exists a factorizationφ : R → R/Rn → A, n >> 0, and therefore the composition φf, φg are homotopy equiv-alent. This means that f and g induces the same morphism between deformation functors[R,−] → [S,−].

Another aspect of prohomotopy is the following

Theorem 4.2. Every complete quasismooth differential algebra is prohomotopy equivalentto a minimal complete quasismooth differential algebra. Two minimal complete quasismoothalgebras are prohomotopy equivalent if and only if they are isomorphic.

Proof As in the nongraded case, a morphism f : S → R of complete quasismooth differen-tial algebras is an isomorphism if and only if f2 : S/S

2 → R/R2 is an isomorphism. Since thecohomology of (S/S2)∨ is isomorphic, up to a shift, to the tangent space of the deformationfunctor [S,−], if f is a prohomotopy equivalence then [S,−] ≃ [R,−] and f2 is a quasiiso-morphism; this proves that every prohomotopy equivalence of minimal complete quasismoothdifferential algebras is an isomorphism.

Let (R, d) be a fixed complete quasismooth differential algebra and let d11 : R/R2 → R/R2

be the differential induced by d. Let R/R2 = H ⊕W be a decomposition with W acycliccomplex and d11(H) = 0; fix homogeneous basis {hj} of H and {vi, wi} of W such thatd11(vi) = wi for every i. Since d(vi) = wi + φi for some φi ∈ R2, up to the analytic change ofcoordinates

hj → hj , vi → vi, wi → dvi

it is not restrictive to assume dW ⊂W . In particular the ideal (W ) ⊂ R is a differential idealand the quotient S = R/(W ) is a minimal complete quasismooth differential algebra.

Our aim is to show that the projection Rπ

−→S is a prohomotopy equivalence; as a firststep we prove that there exists a right inverse γ : S → R and then we prove that γπ : R/Rn →R/Rn is homotopic to the identity for every n > 0.

Since R is complete it is sufficient to find a sequence of morphisms γn : S → R/Rn, n ≥ 2,such that γ2 : S → H = S/S2 → R/R2 = H ⊕W is the natural inclusion and the diagrams

Sγn−→ R/Rn

ց

y

π

S/Sn

Sγn+1

−→ R/Rn+1

y

γn ւ

R/Rn

are commutative.We note that the natural morphism

R/Rn+1 → R/Rn ×S/Sn S/Sn+1

21

is an acyclic small extension for every n ≥ 2. According to 2.3 we may define inductivelyγn+1 : S → R/Rn+1 as a lifting of

(γn, pn+1) : S → R/Rn ×S/Sn S/Sn+1

where pn : S → S/Sn denotes the projection.Up to a change of coordinates of the form hj → γπhj, vi → vi, wi → wi we can assume

that S ⊂ R is the complete subalgebra generated by H . The homotopies

Hn : R → R/Rn[t, dt], hj → hj , vi → vi ⊗ t, wi = dvi → d(vi ⊗ t)

gives the required prohomotopy between γπ and the identity.

Let’s denote by K(quasismooth) the category of quasismooth complete dg-algebras withmorphisms up to prohomotopy equivalence; then there exists the prorepresantability functor

(K(quasismooth))opp → Def , S → [S,−]

Definition 4.3. A deformation functor F : C → Set is called finite if its tangent spaceTF [1] has finite total dimension.

Definition 4.4. A complete quasismooth dg-algebra R is called finite if [R,−] is a finitedeformation functor; i.e. if the graded vector space H∗(R/R

2) has finite total dimension.

Note that a minimal complete quasismooth dg-algebra R is finite if and only if K ⊕ Ris Noetherian. If Def ♭ ⊂ Def denotes the full subcategory of finite deformation functorsand K(quasismooth)♭ ⊂ K(quasismooth) denotes the full subcategory of finite completequasismooth dg-algebras we have

Theorem 4.5. The restriction of the prorepresentability functor

ϑ : (K(quasismooth)♭)opp → Def ♭

is an equivalence of categories.

Proof After theorem 4.2 it is not restrictive to assume that every complete quasismoothdg-algebra is minimal and finite.

Step 1: ϑ is injective on morphisms.Let f, g : S → R be morphisms of dg-algebras inducing the same natural transformation offunctors [R,−] → [S,−]; if πn : R → R/Rn denotes the natural projection then πnf, πng : S →R/Rn are homotopic maps and then f and g are prohomotopic by definition.

Step 2: ϑ is surjective on morphisms.Let S,R be finite complete minimal quasismooth dg-algebras and α : [R,−] → [S,−] be anatural transformation of functors. In the same notation of Step 1 we will construct recursivelya coherent sequence of dg-algebra morphisms γn : S → R/Rn such that [γn] = α[πn] ∈[S,R/Rn]. The resulting inverse limit γ = lim

←γn : S → R will be a morphism of dg-algebras

inducing α.Assume γ1, . . . , γn as above are constructed for a fixed n and let f : S → R/Rn+1 be

a representative of α[πn+1]. Denoting by p : R/Rn+1 → R/Rn the projection there existsa homotopy H : S → R/Rn[t, dt] such that H0 = pf , H1 = γn. As HomDGA(S,−) is apredeformation functor, the argument of Step 3 in the proof of 2.8 shows that there existsa homotopy K : S → R/Rn+1[t, dt] such that pK = H and K0 = f ; it is therefore sufficientdefine γn+1 = K1.

22

Step 3: ϑ is surjective on isomorphism classes.Let F be a deformation functor with finite dimensional tangent space TF [1]. By 3.3 we needto prove that there exists a minimal complete quasismooth dg-algebra (R, d) and a naturaltransformation of functors ξ : [R,−] → F which is an isomorphism on tangent spaces.

For every integer i, let V−i be the dual of the vector space TF [1]i, then V = ⊕iVi =Hom∗(TF [1],K ) is a finite dimensional graded vector space. Thinking of V as an object inDG ∩C, we have by 3.2 a natural isomorphism

F (V ) = ⊕i(TF [1]i ⊗ V−i) = HomDG(TF [1], TF [1]).

We denote by ξ2 ∈ F (V ) the elements corresponding to the identity on TF [1]. Again by 3.2for every I ∈ DG ∩C, the map

ξ2 : [V, I] → F (I), [f ] → f(ξ2)

is bijective. Let R be the inverse limit of R/Rn+1 = ⊕ni=1(⊙

iV ); we want to define twocoherent sequences, the first of square zero differentials dn : R/R

n → R/Rn[1] with d2 = 0and the second of elements ξn ∈ F (R/Rn, dn) lifting ξ2. This will give a structure of minimalcomplete quasismooth dg-algebra on R = limR/Rn and the required natural transformation

[R,−] → F, [f : R → A] → f(ξn), n >> 0.

By induction assume there are defined dn, ξn, since ξn lifts ξ2, by 3.12 and 3.10 there existsa unique square-zero differential dn+1 : R/R

n+1 → R/Rn+1[1] lifting dn and such that theobstruction to lifting ξn to (R/Rn+1, dn+1) vanishes.

Corollary 4.6. For every finite deformation functor F : C → Set, the induced functor[F ] : K(C) → Set is prorepresentable by a quasismooth minimal complete differential al-gebra R. If T iF = 0 for every i ≤ 0 then K ⊕ R is the algebra of functions of a formalpointed noetherian semismooth dg-scheme.

Proof Since prorepresentable means representable by a projective limit the proof comesimmediately from Theorem 4.5.

5 Differential graded coalgebras and L∞-algebras

Definition 5.1. A cocommutative coassociative Z-graded coalgebra is the data of a gradedvector space C = ⊕n∈ZC

n and of a coproduct ∆: C → C ⊗ C such that:

• ∆ is a morphism of graded vector spaces.

• (coassociativity) (∆⊗ IdC)∆ = (IdC ⊗∆)∆: C → C ⊗ C ⊗ C.

• (cocommutativity) T∆ = ∆.

A morphism of coalgebras is a morphism of graded vector spaces that commutes with coprod-ucts.

By coassociativity we can define the iterated coproduct ∆n = (IdC⊗∆n−1)∆: C → ⊗nC.A coassociative coalgebra (C,∆) is called nilpotent if ∆n = 0 for n >> 0.

23

Example 5.2. The reduced symmetric coalgebra is by definition S(V ) = ⊕n>0 ⊙n V , withthe coproduct:

∆(v1 ⊙ . . .⊙ vn) =

n−1∑

r=1

∑

σ∈S(r,n−r)

ǫ(σ)(vσ(1) ⊙ . . .⊙ vσ(r))⊗ (vσ(r+1) ⊙ . . .⊙ vσ(n)).

An easy computation about unshuffles shows that the map N : S(V ) → T (V ) is a morphismof coalgebras.The coalgebra S(V ) is coassociative, cocommutative without counity. Note also that V =ker∆. We shall say that a subcoalgebra C ⊂ S(V ) is homogeneous if C = ⊕n(C ∩ ⊙nV ).

Following [16] we denote by C(V ) the coalgebra (S(V [1]),∆).

Definition 5.3. Let (C,∆) be a coalgebra. A morphism d : C → C[n] of graded vector spacesis called a coderivation of degree n if

∆d = (d⊗ IdC + IdC ⊗ d)∆.

A coderivation d is called a codifferential if d2 = d ◦ d = 0.More generally if θ : C → D is a morphism of coalgebras, a morphism of graded vector spacesd : C → D[n] is called a coderivation (with respect to θ) if

∆Dd = (d⊗ θ + θ ⊗ d)∆C .

Note that, by the rule of sign, if d : C → C[n] is a morphism of graded vector spaces then(IdC ⊗ d)(x ⊗ y) = (−1)nxx ⊗ d(y). The coderivations of degree n of a coalgebra C form avector space denoted by Codern(C,C).

Lemma 5.4. Let θ1, θ2 : C → D be morphisms of coalgebras and assume that there exists agraded subspace B ⊂ C such that ∆C(C) ⊂ B ⊗ B and θ1(b) = θ2(b) for every b ∈ B. Thenevery coderivation C → D[n] with respect to θ1 is also a coderivation with respect to θ2.

Proof Evident.

The reduced symmetric coalgebra is a free object in the the category of graded, locallynilpotent, cocommutative, coassociative coalgebras. In fact holds the following:

Proposition 5.5. Let V be a graded vector space, π : S(V ) → V the projection and C acocommutative coalgebra which is union of nilpotent subcoalgebras.

1. The composition with π gives a bijection between the set of coalgebra morphisms θ : C →S(V ) and morphisms of graded vector spaces m : C → V . The inverse is given by theformula

θ =∞∑

n=1

1

n!mn∆

n−1C ,

where mn(c1 ⊗ . . .⊗ cn) = m(c1)⊙ . . .⊙m(cn).

2. For a fixed coalgebra morphism θ : C → S(V ), the composition with π gives an isomor-phism

Codern(C, S(V )) → HomG(C, V [n]).

24

Proof 1) is proved in [22, Prop. 4.2 on page 285].2) follows easily by 1) in view of the following observation: consider the coalgebra C⊕C[−n]with coproduct

∆(c1 + c2[−n]) = ∆(c1) + ∆(c2)[−n] + T (∆(c2)[−n]).

(In the above formula we used the natural isomorphism (C ⊗ C)[−n] = C[−n] ⊗ C). Thend : C[−n] → S(V ) is a coderivation (with respect to θ) if and only if θ+d : C⊕C[−n] → S(V )is a morphism of coalgebras.

Definition 5.6. By a dg-coalgebra we intend a triple (C,∆, d), where (C,∆) is a gradedcoassociative cocommutative coalgebra and d : C → C[1] is a codifferential of degree 1. Thecategory of dg-coalgebras, where morphisms are morphisms of coalgebras commuting withcodifferentials, is denoted by DGC.

Example 5.7. Given A = ⊕Ai ∈ C let C = A∨ be its graded dual: i.e. C = ⊕Ci, whereCi = HomK (A−i,K ). We denote by <,> : C ×A→ K the induced pairing.The pairing < c1 ⊗ c2, a1 ⊗ a2 >=< c1, a1 >< c2, a2 > gives a natural isomorphism C ⊗C =(A⊗A)∨ commuting with the twisting maps T and we may define ∆ as the transpose of themultiplication map µ : A⊗A→ A.Then (C,∆) is a coassociative cocommutative coalgebra. Note that C is nilpotent if and onlyif A is nilpotent.Defining the codifferential in C as the transpose of the differential in A we get a finitedimensional nilpotent dg-coalgebra.

Definition 5.8. Let V be a graded vector space; a codifferential of degree 1 over the symmet-ric coalgebra C(V ) = S(V [1]) is called an L∞ structure on V . The dg-coalgebra (C(V ), Q)is called an L∞-algebra.A L∞-algebra (C(V ), Q) is called minimal if Q1

1 = 0.

Let (C(V ), Q) be a L∞-algebra and let Qij : ⊙j (V [1]) → ⊙i(V [1])[1] be the composition

of the codifferential Q with the inclusion ⊙j(V [1]) ⊂ C(V ) and the projection C(V )[1] →⊙i(V [1])[1]; by Proposition 5.5 the codifferential Q is uniquely determined by the linear mapsQ1

j , j > 0 by the explicit formula (cf. [21])

Q(v1 ⊙ . . .⊙ vn) =

n∑

k=1

∑

σ∈S(k,n−k)

ǫ(σ)Q1k(vσ(1) ⊙ . . .⊙ vσ(k))⊙ vσ(k+1) ⊙ . . .⊙ vσ(n).

In particular we have Qij = 0 for every i > j and then the subcoalgebras ⊕r

i=1(⊙iV ) are

preserved by Q.We define a functor MCV : C → Set by setting:

MCV (A) = HomDGC(A∨, C(V ))

We use the following terminology: for every coalgebra C and every linear map φ : C →C(V )[n] we denote by φi : C → ⊙i(V [1])[n] the composition of φ with the projectionC(V )[n] →⊙i(V [1])[n].

Proposition 5.9. For every L∞-algebra (C(V ), Q), the functor MCV is a predeformationfunctor.

25

Proof It is evident that MCV is a covariant functor and MCV (0) = 0. Let α : A → C,β : B → C be morphisms in C, then it is easy to see that the dg-coalgebra (A×C B)∨ is thefibered coproduct, in the category DGC, of the morphisms

C∨tα−→ A∨

y

tβ

B∨

and then

MCV (A×C B) =MCV (A)×MCV (C) MCV (B).

Let 0−→I−→A−→B−→0 be a small acyclic extension inC, we want to prove thatMCV (A) →MCV (B) is surjective.We have a dual exact sequence

0−→B∨−→A∨−→I∨−→0, B∨ = I⊥.

Since IA = 0 we have ∆A∨(A∨) ⊂ B∨ ⊗B∨.Let φ ∈ MCV (B) be a fixed element and φ1 : B∨ → V [1]; by Proposition 5.5 φ is uniquelydetermined by φ1. Let ψ1 : A∨ → V [1] be an extension of φ1, then, by 5.5, ψ1 is induced bya unique morphism of coalgebras ψ : A∨ → C(V ).The map ψdA∨ −Qψ : A∨ → C(V )[1] is a coderivation and then, setting h = (ψdI∨ −Qψ)1 ∈HomG(I∨, V [2]), we have that ψ is a morphism of dg-coalgebras if and only if h = 0.Note that ψ1 is defined up to elements ofHomG(I∨, V [1]) = (V [1]⊗I)0 and, since ∆A∨(A∨) ⊂B∨ ⊗B∨, ψi depends only by φ for every i > 1. Since I is acyclic and hdI∨ +Q1

1h = 0 thereexists ξ ∈ HomG(I∨, V [1]) such that h = ξdI∨ − Q1

1ξ and then θ1 = ψ1 − ξ induces adg-coalgebra morphism θ : A∨ → C(V ) extending φ.

Definition 5.10. Let (C(V ), Q) be a L∞-algebra and let DefV = MC+V be the deformation

functor associated to the predeformation functor MCV (Theorem 2.8). We shall call DefVthe deformation functor associated to the L∞-algebra (C(V ), Q).

A morphism of L∞-algebras C(V ) → C(W ) induces in the obvious way a natural trans-formation MCV →MCW and then, by Theorem 2.8, a morphism DefV → DefW .

For every A ∈ C there exists an algebra structure on HomG(A∨, C(V )) induced by thenatural isomorphism HomG(A∨, C(V )) = (C(V ) ⊗ A)0; the product of f, g : A∨ → C(V ) isgiven by fg = µ(f ⊗ g)∆A∨ , where µ : C(V ) ⊗ C(V ) → C(V ) is the multiplication of thereduced symmetric algebra C(V ) = S(V [1]). According to 5.5 a morphism f : A∨ → C(V ) isa morphism of coalgebras if and only if there exist m ∈ HomG(A∨, V [1]) such that

f = exp(m)− 1 =

∞∑

n=1

1

n!mn.

Lemma 5.11. Let Q be a L∞ structure on V and m ∈ (V [1]⊗A)0. Then exp(m)−1 belongsto MCV (A) = HomDGC(A

∨, C(V )), via the isomorphism HomG(A∨, C(V )) = (C(V )⊗A)0,if and only if

IdV [1] ⊗ dA(m) = Q1 ⊗ IdA(exp(m)− 1) =

∞∑

n=1

1

n!Q1

n ⊗ IdA(mn).

26

Proof Denote θ = exp(m) − 1: A∨ → C(V ), then θdA∨ , Qθ are coderivations and thenθdA∨ = Qθ if and only if

IdV [1] ⊗ dA(m) = (θdA∨)1 = Q1θ = Q1 ⊗ IdA(exp(m)− 1).

The above lemma allow to define MCV (A) for every nilpotent dg-algebra A; therefore wecan define, as in [16, 4.5.2], the deformation functor DefV as the quotient of MCV by theequivalence relation given by: x, y ∈ MCV (A), x ∼ y if and only if there exists a solutionz ∈ MCV (A[t, dt]) such that x = zt=0, y = zt=1. The equivalence of this definition withMC+

V follows immediately from the equality A[t, dt] = ∪ǫ>0V ⊗A[t, dt]ǫ, which implies thatMCV (A[t, dt]) is the direct limit of MCV (A[t, dt]ǫ), and the explicit construction of MC+

V

given in Theorem 2.8.

Lemma 5.12. Let (C(V ), Q) be a L∞-algebra and consider V as a differential graded vectorspace with differential Q1

1. Then for every A ∈ C ∩DG we have

1. MCV (A) = HomDG(A∨, V [1]) = Z1(V ⊗A),

2. DefV (A) = H1(V ⊗A) and then T iDefV = Hi(V ).

Proof Since A2 = 0 we have ∆A∨ = 0 and then φi = 0 for every i ≥ 2 and everymorphism of coalgebras φ : A∨ → C(V ): this proves 1). The proof of 2) follows from 1) andLemma 2.10.

The following result is well known (cf. [16], [18]):

Proposition 5.13. Given a graded vector space V , there exists a bijection between the set ofDGLA structures (V, d, [, ]) and the set of L∞-algebra structures (C(V ), Q) such that Q1

j = 0for every j ≥ 3.Explicitly the bijection is given by d(w)[1] = −Q1

1(w[1]) and [w1, w2] = (−1)w1Q12(w1[1] ⊙

w2[1])

An easy computation shows that for every DGLA (V, d, [, ]) the definitions 2.16 and 5.10give the same functor MCV , cf. [16].

6 The Lie algebra structure on TF

Let F be a deformation functor, T = TF . For every finite dimensional graded subspaceV ⊂ T [1] we have F (V ∨) = HomG(V, T [1]) and therefore there exists a distinguished elementξV ∈ F (V ∨) corresponding to the inclusion V ⊂ T [1].

Proposition 6.1. There exists a unique morphism of graded vector spaces Q12 : ⊙2 (T [1]) →

T [2] with the following property:For every pair of finite dimensional graded subspaces V ⊂ H ⊂ T [1] such that Q1

2(⊙2V ) ⊂ H

consider the subcoalgebra A = H ⊕ (⊙2V ) ⊂ C(T ) endowed with the codifferential

(

0 Q12

0 0

)

: H⊕(⊙2V ) → (H⊕(⊙2V ))[1].

Then ξH lifts to F (A∨).

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Proof Let V ⊂ T [1], taking the dual of the exact sequence

0−→V−→V⊕(⊙2V )−→⊙2V−→0

we get a small extension of finite dimensional nilpotent graded algebras

0−→(⊙2V )∨−→V ∨⊕(⊙2V )∨−→V ∨−→0.

Let obV : F (V ∨) = HomG(V, T [1]) → F ((⊙2V )∨[1]) = HomG(⊙2V, T [2]) be the associatedobstruction map and let qV = obV (ξV ). Note that qV is, up to sign, the primary obstructionmap introduced in 3.9. By base change, for every V ⊂ H ⊂ T [1], qV is the obstruction tolifting ξH to F (H∨⊕ (⊙2V )∨). In particular if the image of qV (⊙2V ) ⊂ H [1], by Lemma 3.10the codifferential

(

0 −qV0 0

)

: H⊕(⊙2V ) → (H⊕(⊙2V ))[1]

is the unique codifferential in A = H ⊕ ⊙2V such that ξH lifts to F (A∨). Glueing all the−qV , where V range over all finite dimensional graded subspaces of T [1], we get the requiredmap Q1

2.

Theorem 6.2. Let Q : C(T ) → C(T )[1] be the linear map defined by

Q(v1 ⊙ . . .⊙ vn) =∑

σ∈S(2,n−2)

ǫ(σ)Q12(vσ(1) ⊙ vσ(2))⊙ vσ(3) ⊙ . . .⊙ vσ(n).

Then Q is a codifferential and then (C(T ), Q) is a minimal L∞ algebra.

Proof According to standard formula for L∞ algebras, see e.g. [21], the mapQ is a coderiva-tion of degree 1 and Q2 = 0 if and only if Q2(⊙3T [1]) = 0.Let V ⊂ W ⊂ Z ⊂ T [1] be finite dimensional subspaces such that Q(⊙2V ) ⊂ W [1] andQ(⊙2W ) ⊂ Z[1]. Then B = Z ⊕ (⊙2W ) ⊕ (⊙3V ) is a subcoalgebra of C(T ) such thatQ(B) ⊂ B. Let δ : (Z ⊕ (⊙2W ))∨ → (⊙3V )∨[2] be the transpose of Q2, by Proposition 3.12we have δF ((Z ⊕⊙2W )∨) = 0. Since the element ξZ lifts to some ξ ∈ F ((Z ⊕ (⊙2W ))∨) andδ(ξ) = 0 we also have that δ = 0 and then also Q2(B) = 0.The L∞ algebra (C(T ), Q) is clearly minimal and, since Q1

j = 0 for every j ≥ 3 it comesfrom a graded Lie algebra (T, [, ]).

7 Geometric deformation functors

In this section we are interested, given a deformation functor F , to find sufficient conditionsfor the existence of a L∞-algebra V such that F = DefV . A straightforward generalizationof the transpose of the proof of Corollary 4.6 in the set-up of coalgebras shows that, if T iF isfinite dimensional for every i, then there exists a structure of minimal L∞-algebra on T = TFsuch that F = DefT .

Here we consider a special class of functors that, in our opinion, contains all the “inter-esting” deformation functors.

By extending the analogue definition for classical deformation functors [6] we shall saythat a predeformation functor G is homogeneous if for every pair of morphisms A → C,B → C in the category C we have

G(A×C B) = G(A) ×G(C) G(B).

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We say that a deformation functor F if geometric if F = G+ for some homogeneous prede-formation functor G. It is not clear to us whether every deformation functor is geometric; inany case we strongly suspect that all the concrete example of deformation functors havingsome algebro-geometric interest have this property. All the examples of deformation functorsconsidered in this paper are geometric.

Theorem 7.1. Let F be a geometric deformation functor and T = TF . Then there exists aminimal L∞ structure on T and an isomorphism of functors ξ : DefT → F .

Before proving the theorem we need some preparatory results.

Definition 7.2. Let (C(V ), Q) be a L∞ algebra and {vi} a fixed homogeneous basis of V [1].We shall say that a dg-subcoalgebra A ⊂ C(V ) is standard if A has a basis of monomialsvi1 ⊙ . . .⊙ vir ; in particular every standard dg-subcoalgebra is homogeneous.

Lemma 7.3. Let (C(V ), Q) be a minimal L∞-algebra, then it is the union of its finite di-mensional standard dg-subcoalgebras (with respect to a fixed homogeneous basis of V [1]).

Proof For every finite dimensional subspace W ⊂ C(V ) there exist n > 0 and a finitedimensional standard subspace Hn such that W ⊂ ⊕n

i=1(⊙iHn). For every sequence of finite

dimensional standard subspaces

0 ⊂ Hn ⊂ Hn−1 ⊂ . . . ⊂ H1 ⊂ V [1]

we can consider the standard subcoalgebra C = ⊕ni=1(⊙

iHi). The condition Q(C) ⊂ C isequivalent to

Q1j(⊙

jHs) ⊂ Hs+1−j , ∀s ≥ j ≥ 1.

Assume we have the subspaces Hn, . . . , Hr satisfying the above conditions. If r > 1 then wemay take Hr−1 any finite dimensional standard subspace containing

∑

j≥2Q1j(⊙

jHr+j−2).

Lemma 7.4. Let (C(V ), Q) be a fixed minimal L∞-algebra and {vi} a fixed homogeneousbasis of V [1]; denote by Cn the dg-subcoalgebra ⊕n

i=1 ⊙i (V [1]). For a fixed integer n ≥ 2 let

A be the set of finite dimensional standard dg-subcoalgebras of Cn; A is a directed set withthe ordering given by inclusions. Let f : A → Set be a contravariant functor such that:

1. if A ⊂ Cn−1, then f(A) = {∗} is the one-point set.

2. f(A+B) → f(A)×f(A∩B) f(B) is bijective.

3. f(A) 6= ∅ for every A ∈ A.

Then

f(A) := liminvA∈A

f(A) 6= ∅.

Proof We shall say that a subset Λ ⊂ A is saturated if:

1. If A ⊂ Cn−1 then A ∈ Λ.

2. If A,B ∈ A, A ⊂ B and B ∈ Λ then A ∈ Λ.

3. If A,B ∈ Λ then A+B ∈ Λ.

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For every saturated Λ let f(Λ) be the set of coherent sequences x ∈∏

A∈Λ f(A).Let B the set of all pairs (Λ, x) with Λ ⊂ A saturated, f(Λ) 6= ∅ and x ∈ f(Λ). On B thereexists a natural ordering: (Λ1, x1) ≤ (Λ2, x2) if and only if Λ1 ⊂ Λ2 and x2 extends x1.Clearly B 6= ∅ and every chain in B has an upper bound. By Zorn’s lemma there exists amaximal element (Λ, x) ∈ B; we want to prove that Λ = A.Assume Λ 6= A and let A ∈ A−Λ such that the dimension of A is minimum. Necessarily wemust have dimK (A) = dimK (A∩Cn−1) + 1, A∩B ⊂ Cn−1 and f(A+B) = f(A)× f(B) forevery B ∈ Λ. Let Λ′ be the union of Λ and {A+B |B ∈ Λ}. Choose an element xA ∈ f(A),then it is uniquely determined an element xA+B ∈ f(A + B) for every B ∈ Λ. We need toprove that, if C ⊂ A+B then either C = A+B or C ∈ Λ′ and xA+B extends xC .Assume C 6= A + B, since A,B,C are standard we have C = (A ∩ C) + (B ∩ C). If A ⊂ Cthen C = A + (B ∩ C) ∈ Λ′, since xB extends xB∩C also xA+B extends xC . If A ∩ C 6= Athen C ∩A ∈ Λ, C ∩A ⊂ Cn−1 and f(C) = f(C ∩B); since xA+B extends xB it also extendsxC .

Remark 7.5. It is clear that Lemma 7.4 also works if the codifferential Q is defined only inthe subcoalgebra Cn.

Proof of 7.1 Denote by Sn = ⊕ni=1(⊙

i(T [1])). We recall that for every graded vector spaceI ∈ C ∩G we have

F (I) = HomG(I∨, T [1]) = HomDGC(I∨, S1),

where S1 is endowed with the codifferential d1 = 0. In particular for every finite dimensionalstandard subspace J ⊂ T [1] we have F (J∨) = HomG(J, T [1]): we denote by ξJ ∈ F (J∨) theelement corresponding to the inclusion J → T [1] and by

ξ1 = (ξJ ) ∈∏

J⊂T [1]

F (J∨).

Let G be a homogeneous predeformation functor such that F = G+ and let π : G → F theprojection.We now construct recursively a sequence (dn, ηn) such that for every n ≥ 1

An: dn : Sn → Sn−1[1] ⊂ Sn[1] is a codifferential extending dn−1.

Bn: ηn ∈ liminv F (G∨), where:

1. the limit is taken over all the finite dimensional standard dg-subcoalgebras of(Sn, dn).

2. η1 is a lifting of ξ1.

3. ηn is a lifting of ηn−1 if n ≥ 2.

For every standard subspace J ⊂ T [1] let f(J) ⊂ G(J∨) be the set of liftings of ξJ ; thefunctor f satisfies the hypothesis of Lemma 7.4 and then there exists η1 that lifts ξ1. Letn ≥ 2 be a fixed integer and assume are given (d1, η1), . . . , (dn−1, ηn−1) satisfying Ai, Bi,i = 1, . . . , n− 1.Let A ⊂ Sn be a fixed standard finite dimensional subcoalgebra and denote B = A ∩ Sn−1,J = A ∩ S1. Possibly enlarging B, we may assume that dn−1(B) ⊂ B and dn−1 can beextended to a coderivation on A; dualizing we get a small extension of finite dimensionalgraded nilpotent algebras

0−→B⊥−→A∨−→B∨−→0,

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such that B⊥ and B∨ have respective differential equal to 0 and tdn−1.Let ξB = π(ηB) ∈ F (B∨) be the projection of ξn−1 = π(ηn−1). Since ξB lifts ξJ andJ∨ = B∨/(B∨)2 the map

[B∨, B⊥[2]] → F (B⊥[2]), [f ] → f(ξB)

is injective and, by Proposition 3.12, there exists a differential on A∨ making the exactsequence 0 → B⊥ → A∨ → B → 0 a small extension in C.Let o ∈ F (B⊥[1]) = HomG(A/B, T [2]) be the obstruction to the lifting of ξB to F (A∨);possibly enlarging B, we may suppose that o ∈ HomG(A/B, J [1]) and then there exists anunique morphism φ : B∨ → B⊥[1] such that o = φ(ξB). By Lemma 3.10 there exists anunique differential on A making 0 → B⊥ → A∨ → B → 0 a small extension in C and suchthat ξB lifts to F (A∨). The transpose of this differential is a codifferential in A.Clearly all these codifferentials in the finite dimensional homogeneous subcoalgebras of Sn

can be glued together and we get a global codifferential dn : Sn → Sn−1[1].For every A ⊂ Sn finite dimensional standard dg-subcoalgebra let

f(A) = {x ∈ G(A∨) |x is a lifting of ηA∩Sn−1}.

By Lemma 2.9 and the choice of dn, f(A) 6= ∅ for every A. The functor f satisfies thehypothesis of Lemma 7.4 and then liminv f(A) 6= ∅. By construction every ηn ∈ liminv f(A)satisfies the condition Bn.The sequence η = (ηn) gives a natural transformation ξ : MCT → G by the rule:

HomDGC(A∨, C(T )) ∋ α→ η(f) = tα(ηD) ∈ F (A),

where D is a finite dimensional standard dg-subcoalgebra of C(T ) such that α(A∨) ⊂ D.By construction the induced morphism DefT = MC+

T → F is an isomorphism on tangentspaces and then, according to Corollary 3.3 it is an isomorphism of functors.

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Marco ManettiDipartimento di Matematica “G. Castelnuovo”,Universita di Roma “La Sapienza”,Piazzale Aldo Moro 5, I-00185 Roma, Italy.manetti@mat.uniroma1.it, http://www.mat.uniroma1.it/people/manetti/

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